Move tear out tabs between Chrome windows in AwesomeWM
Your friend will awful.client.floating.toggle function. You can bind it to a hotkey, in the default configuration is modkey+ctrl+space, check source code.
With default settings you can set float property (and behavior) of a client with press modkey+ctrl+space, and after you can move over the main chrome-window.
This appears to be a bug in Chrome or Awesome at this time. This problem has been around for a while now: https://bbs.archlinux.org/viewtopic.php?id=159628
The problem only happens on one of my monitors for me. I submitted the issue to Google, but chances are they may not fix it anytime soon. You may want to consider using another browser or window manager until this issue is fixed.